
with \(M_* = \left[90/\left(\pi ^2 g_*\right)\right]^{1/2} M_{\rm Pl}\) , \(\rho _{\rm tot} = 3H^2M_{\rm Pl}^2\) , and where \(\rho \)  and \(\rho _B = B^2/\left(2a^4\right)\)  denote the energy densities of the plasma and hypermagnetic field, respectively.
In the first case, we assumed that the energy in the velocity field saturates the initial energy in the hypermagnetic field, which acts as a source term for the velocity field.
This may be regarded as an optimistic estimate that minimizes the parameter region where \(\textrm {Re}_{\rm mag}^{\rm ini} \lnot \gg 1\) .
Meanwhile, in the second case, we assumed that a viscous regime is reached, where the kinetic viscosity is balanced by the source term from the hypermagnetic field.
For more details on these analytical estimates, see Ref. [1]}.
Besides, we stress that a more refined analysis of the onset of turbulence after axion inflation requires a dedicated MHD analysis, which has thus far not been performed in the regime of interest, \(\textrm {Re}_{\rm mag} \gg \textrm {Re}_{\rm kin}\) .
